Question : If $\sin A=\frac{2}{5}$, where $A$ is an acute angle, what is the value of $\frac{5 \sin A+2 \operatorname{cosec} A}{\sqrt{21} \sec A}$?
Option 1: $\frac{5}{4}$
Option 2: $\frac{7}{5}$
Option 3: $\frac{4}{5}$
Option 4: $\frac{5}{7}$
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Correct Answer: $\frac{7}{5}$
Solution : $\sin A=\frac{2}{5}$ $⇒\cos A = \sqrt{1-\sin^2 A}$ $⇒\cos A = \sqrt{1-(\frac{2}{5})^2}$ $⇒\cos A = \sqrt{1-(\frac{4}{25})}$ $⇒\cos A = \sqrt{(\frac{21}{25})}$ $⇒\cos A = \frac{\sqrt{21}}{5}$ So, $\sec A = \frac{5}{\sqrt{21}}$ And, $\operatorname{cosec} A = \frac{5}{2}$ $\frac{5 \sin A+2 \operatorname{cosec} A}{\sqrt{21} \sec A}$ $=\frac{5 \times \frac{2}{5} +2 \times \frac{5}{2} }{\sqrt{21} \times \frac{5}{\sqrt{21}}}$ $=\frac{2 +5}{5}$ $=\frac{7}{5}$ Hence, the correct answer is $\frac{7}{5}$.
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